Isomorphisms Of Finite Bi-Cayley Graphs

In algebraic graph theory, the study of isomorphisms of graphs has a long history. In this thesis, we investigate the isomorphism problem for bi-Cayley graphs and also the structures of Sylow subgroups of BCI-groups.A bipartite graph is a graph which has no odd length cycles. The family of bipartite graphs plays an important role in graph theory. In particular, a bi-Cayley graph Γ is a bipartite graph with the following property:there exists a subgroup of the automorphism group Aut(Γ) which is regular on the two biparts of the vertex set of Γ. In fact, bi-Cayley graphs can also be constructed from groups directly. Let G be a finite group, S be a subset of G (probably contains the identity). Then the bi-Cayley graph of G with respect to S is the bipartite graph with vertex set G x{0,1} and edge set{(g,0),(sg,1)|g∈G,s∈S}, denoted by BCay(G,S).For a finite group G and a subset S(?)G\{1}, we can define the well-known Cayley graph Γ as the graph with vertex set G and arc set{(g, sg)|g∈G,s∈S}, denoted by Cay(G, S). For Cayley graphs and bi-Cayley graphs, there are con-nections between Cay(G, S) and BCay(G, S). For example, BCay(G, S) is the standard double cover of Cay(G, S). Meanwhile, there are many differences be-tween Cay(G, S) and BCay(G, S). For example, BCay(G, S) is always undirected but Cay(G, S) is undirected if and only if S=S-1; Cay(G, S) is vertex transitive, but BCay(G, S) may be not vertex transitive, see.The isomorphism problem of Cayley graphs is called the CI property of Cayley graphs. It has been studied extensively. However, the study of the iso- morphism problem of bi-Cayley graphs is a new topic, and there are limit results about it until now, see. Thus it is an interesting question to study the iso-morphism problem of bi-Cayley graphs. Analogous to the CI property of Cayley graphs, we can define the BCI property of bi-Cayley graphs, see definition2.6.In, it has been proved that every finite group G is a1-BCI-group; G is a2-BCI-groups if and only if for any pair of same order elements a, b, there exists a∈Aut(G) such that a(?)=b or b-1. Thus in this thesis, we will mainly study m-BCI-groups for m≥3.We know that the understanding of the structures of Sylow subgroups of groups are very important for the understanding of the groups themselves, and so we first determine all the possibilities of the Sylow subgroups of3-BCI-groups. We show that for a3-BCI-group G, its Sylow2-subgroup is elementary abelian, cyclic, or the quaternion group Q8; its Sylow p-subgroup is homocyclic where p is an odd prime.Secondly, as an application of the above result, we decide all non-abelian simple3-BCI-groups. We prove that a non-abelian simple group G is a3-BCI-group if and only if G=A5.Thirdly, we investigate the m-BCI property for finite cyclic groups where m≥3, in particular, for cyclic groups of order2p and pn where p is a prime and n is a positive integer. We show that cyclic groups of order2p are3-BCI-groups and cyclic groups of order pn are (p-1)-BCI-groups.Finally, we determine the BCI property of small order groups. It is shown that except for D8, Z8and Z4×Z2, all groups of order less than9are BCI-groups.